# Equivalence Principle

## The Equivalence Principle

### The Newtonian Version

Gravitational mass is the charge to which gravity couples. Inertial mass is a measure of how fast an object accelerates--given the same force, increasing the inertial mass implies decreasing acceleration. The simplest way to state the equivalence principle is this: inertial mass and gravitational mass are the same thing. Then, gravitational force is proportional to inertial mass, and the proportionality is independent of the kind of matter. This implies the Universality of Free Fall(UFF): in a uniform gravitational field, all objects fall with the same acceleration, e.g. 9.8m/s2 near the surface of the earth.

### The Einsteinian Version

All objects fall the same way under the influence of gravity; therefore, locally, one cannot tell the difference between an accelerated frame and an unaccelerated frame. Consider the famous example of a person in a falling elevator. The person floats in the middle of an elevator that is falling down a shaft. Locally, that is during any sufficiently small amount of time or over a sufficiently small space, the person falling in the elevator can make no distinction between being in the falling elevator or being in completely empty space, where there is no gravity.

We could imagine two apples floating on either side of the person; as the elevator approached the earth, the apples would approach each other. This happens because their paths, both toward the center of the earth, eventually converge. But this is not an effect that can be detected in a local experiment.

This statement of the equivalence principle makes an important suggestion. In special relativity--and all classical mechanics--we are used to the idea that objects travel at constant velocity unless a force acts on them. Now, if we can't locally tell the difference between falling in a gravitational field and travelling at constant velocity, then, locally, they must be the same thing. The paths of free bodies define what we mean by "straight" and if we observe an object deviate from constant velocity, it must be because spacetime itself is curved.

Formally, we state the equivalence principle this way: in any and every locally Lorentz (inertial) frame, the laws of special relativity must hold. From this, we conclude that the only things which can define the geometric structure of spacetime are the paths of free bodies.

### The Strong and Weak Equivalence Principles

Often, one finds references to the "strong" or "weak" equivalence principle. The weak equivalence principle has been stated, in the equality of gravitational and inertial mass and in the statement about special relativistic laws holding in every locally Lorentz frame, if we restrict that statement to the "laws of freely falling bodies." The strong equivalence principle applies to all laws of nature, and implies that even gravitational self-energy must obey the equivalence principle.

### What measurements have we made?

We have tested the Equivalence Principle (more precisely the Universality of Free Fall or UFF) for the following cases:

1. Be, Al, Cu and Si test bodies using the Earth as attractor. Our null results consitute the most precise laboratory tests of the UFF. Our results showed that the proposed "fifth force" and its natural generalizations did not exist.

2. Be, Al, Cu and Si test bodies attracted toward the Sun. The goal here was to complement the lunar laser-ranging data that, in effect, tests the UFF for the Earth and Moon falling toward the Sun. The laser-ranging data are used to test whether gravitational self-energy obeys the Equivalence Principle, because the Earth's mass has a 4 × 10-10 contribution from gravitational binding energy but the contribution to the Moon's mass is only 2 × 10-11. However, the laser-ranging test probes a combination of two effects, the differing gravitational self-energies and also the differing compositions (the Earth has an Fe-Ni core while the moon does not) of the Earth and Moon. We are doing an experiment where we compare the accelerations toward the sun of special test bodies that have compositions very close to that of the Earth's core and to that of the Moon's surface. We expect to do this well enough to take full advantage of the precision of the laser-ranging data.

3. Be, Al, Cu and Si test bodies attracted toward the center of our Galaxy. The idea here was to test { in the laboratory} whether the only significant force between dark matter and ordinary matter is gravitational (i.e. obeys the UFF). Our work showed that for Galactic dark matter this was indeed the case. We are developing the Eöt-Wash III instrument to extend our test to cosmological dark matter; because the relevant accelerations are smaller, we need better experimental sensitivity.

4. Cu and Pb attracted toward a compact 3-ton 238U attractor. The goal here was to test the UFF at short ranges, to use an attractor with a very different N/Z ratio than the earth, and to close the "gap" at ranges between 10 km and 1000 km where Eötvös-type experiments using the earth attractor have little sensitivity. Uranium was used because of its high density; this allowed us to place a substantial mass close to the torsion balance. This instrument provided useful constraints for ranges down to 1 cm, i.e. exchange-boson masses up to 2 × 10-5, covering the "Turner window" where astrophysical constraints are very weak.

## Equivalence Principle Violations

The equivalence principle (EP) states that all laws of special relativity hold locally, regardless of the kind of matter involved, and for a long time there was no reason to doubt this. Modern quantum theories, however, often require that at some scale the EP must be violated. Usually, the scale is small, no more than a few centimeters.

What sorts of forces violate the equivalence principle? In a way, one type of EP violation is familiar: any vector field which couples to a mass must violate the equivalence principle. To see this, consider electromagnetism, which is a vector field. There are two electrical charges; a positive charge behaves quite differently from a negative charge in an electric field. The existance of a charge and an anticharge is a general feature of vector fields. Then, if a vector field coupled to mass, there would have to be a mass and an antimass which would behave oppositely in the same gravitational field and therefore violate the EP.

Scalar fields also produce EP violations. Scalar charges, unlike vector charges, are not conserved. The statement of charge conservation for a vector charge is Lorentz invariant. The charge of an object is the integral of the time component of its vector current density, which picks up a factor under Lorentz transformation, over a volume element, which picks up a factor 1/. Therefore, the integral as a whole is Lorentz invariant. For a scalar charge, the relevant integral is a charge density (a Lorentz scalar) integrated over a volume; only the volume picks up a factor under a Lorentz transformation, so scalar charges are not conserved and depend on .

Now, strange things can happen because of that factor. Quarks inside the protons and neutrons are highly relativistic; electrons surrounding the nucleus move more slowly (v/c ~ Z ). We therefore expect scalar interactions to be composition dependent, since larger atoms' electrons are much farther away from the nucleus and move much more slowly.]